Note on Two-f-Tolerance Competition Graphs

Let φ be a symmetric function defined from N×N into N, where N denotes the nonnegative integers. A graph G=(V,E) is a φ-tolerance competition graph if there is a digraph D=(V,A) such that each vertex v
_i
∈V can be assigned a nonnegative integer t
_i
such that, for i≠j, v
_i
v
_j
∈E if and only if |O(v
_i
)∩O(v
_j
)|≥φ(t
_i
,t
_j
). Brigham et al. defined the two-φ-tolerance competition graph as a tolerance competition graph in which all the t
_i
are selected from a 2-set. They characterized such graphs and discussed the relationships between them for φ equal to the minimum, maximum, and sum functions, with emphasis on the situation in which the 2-set is {0,q}. In this paper, we continue to study two-φ-tolerance competition graphs. Characterizations of such graphs, and the lower bounds of θ
_φ
^{p,q}
(G), are presented for φ=min, max and sum, respectively, with emphasis on the situation in which the 2-set is {p,q}.